Melanosis: This condition is characterized by an abnormal deposition of melanin in the tissues, which could explain the brown discoloration under the cornea and on the sclera. It's a relatively common condition that affects the pigmentation of the eyes.

Scleral Melanocytosis: A benign condition where there's an increase in melanin production in the sclera, leading to discoloration. It's more common in certain ethnic groups and can cause brown pigmentation.
Corneal Pigmentation: Various conditions can lead to pigmentation on the cornea, including injuries, infections, or systemic diseases. The pigmentation can appear as brown spots or patches under the cornea.
Ochronosis: A condition caused by the accumulation of homogentisic acid in the body, leading to pigmentation in connective tissues, including the sclera and cornea. It can cause a brown or blue-black discoloration.

Malignant Melanoma: Although less likely, it's crucial to rule out melanoma, especially if there's a new or changing pigmented lesion. Melanoma of the eye can present with pigmentation under the cornea or on the sclera and is potentially life-threatening if not diagnosed early.
Wilson's Disease: A genetic disorder that leads to copper accumulation in the body, including the eyes. It can cause a Kayser-Fleischer ring, which is a brownish discoloration around the cornea due to copper deposition.

Alkaptonuria: A rare genetic disorder that affects the breakdown of certain amino acids, leading to the accumulation of homogentisic acid and causing ochronosis. It's characterized by a blue-black pigmentation in connective tissues, including the eyes.
Primary Acquired Melanosis (PAM): A rare condition that involves the benign or malignant proliferation of melanocytes in the conjunctiva or cornea, leading to pigmentation. It's essential to differentiate PAM from other causes of pigmentation due to its potential for malignant transformation.